Various tire constructions have been developed which enable a tire to run in an uninflated or underinflated condition. Non-pneumatic tires do not require inflation, while “run flat tires” may continue to operate after receiving a puncture and a complete or partial loss of pressurized air, for extended periods of time and at relatively high speeds. One such type of run flat tire incorporates a thin annular high strength band element which acts as a tension member when the tire is pressurized and acts as a structural compression member when the tire is in the unpressurized or partially pressurized state.